What is the inverse of the function $f(x)=\dfrac{6x-5}{x+9}$ ? $ f^{-1}(x) =$
Solution: Let's start by replacing $f(x)$ with $y$. $y=\dfrac{6x-5}{x+9}$ Now let's swap $x$ and $y$ and solve for $y$. $\dfrac{6y-5}{y+9}=x$ [Why do we swap x and y?] $\begin{aligned} \dfrac{6y-5}{y+9}&=x \\\\ 6y-5&=x(y+9) \\\\ 6y-5&=xy+9x \\\\ 6y-xy&=9x+5 \\\\ y(6-x)&=9x+5 \\\\ y&=\dfrac{9x+5}{6-x} \end{aligned}$ In conclusion, this is the inverse function: $f^{-1}(x)=\dfrac{9x+5}{6-x}$ [I saw someone solve this problem by originally solving for x. Were they wrong?]